Due to the presence of the \(\Phi \) -Laplacian the suitable function space to seek weak solutions of Problem \((P)\)  is the Orlicz-Sobolev space \(W^{1,\Phi }(\Omega )\) . We recall below the definition and some basic properties of the Orlicz and Orlicz-Sobolev spaces that will be used to derive a variational formulation for our problem, then to prove the existence of at least one weak solution. For more details we refer to [1]}, [2]}, [3]}, [4]}.
